show that the given equation is a solution of the given differential equation

xy' + y^2 = 0 , xy = cx + cy

Please help me solve this.

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Are you sure that these equations are the given equations? Because they are not dimensionally consistent. The right hand equation requires that c, x, and y all have the same units (call them 'fred'). This means that y'=dy/dx is dimensionless. This in turn means that the left hand equation has units of 'fred' for the first term and 'fred'^2 for the second term.

I have no choice but to assume that 'c' is a constant. But what do you mean "as if *they* were constants," there's only one 'c'. If you mean to say that the two occurances are not the same constant, then you need to call them something else.

And hopefully the OP (and others?) will take note that I didn't even have to try to solve the problem to spot (1) that the given equation could not be a solution of the given differential equation, and that (2) what the form of the diffy-Q would need to be in order for it to be a solution. That's part of the power of dimensional analysis -- even when there are no dimensions, per se.

Actually, the problem isn't pure algebra, since it involves a differential equation. But if you treat it as a "pure math problem", then you have little choice but to crank through the math and solve for y, take the derivative, square y, and plug all of that into the differential equation. Then, when you discover that it doesn't work, you are left to wonder whether it really doesn't work, or whether you just made a mistake along the way. So now you spend more time redoing it and/or checking your work to make sure that you didn't make any mistakes. For a simple problem like this, all of that doesn't take too long. But this could easily have been something that took a couple pages to work out (without looking much more complicated at face value than this one). But in either case, if you look for dimensional consistency, even in a 'pure math problem', you can spot things like this without doing any math at all and KNOW that it won't work out.

Perhaps I'm just silly, but I prefer to spot problems early whenever possible.

I totally agree with WBahn. I teach my students to use dimensional analysis whenever possible, especially to do a quick check of answers for reasonableness. By the way, I teach dimensional analysis to my honors Algebra I students in September so they have yet one more tool in their toolbox. As students mature in their math knowledge, the applications are endless.